Properties

Label 1-6025-6025.1809-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.249 + 0.968i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.669 + 0.743i)2-s + (0.5 − 0.866i)3-s + (−0.104 − 0.994i)4-s + (0.309 + 0.951i)6-s + (−0.978 + 0.207i)7-s + (0.809 + 0.587i)8-s + (−0.5 − 0.866i)9-s + (−0.669 + 0.743i)11-s + (−0.913 − 0.406i)12-s + (−0.104 − 0.994i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)16-s + (−0.809 + 0.587i)17-s + (0.978 + 0.207i)18-s + (0.104 − 0.994i)19-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (0.5 − 0.866i)3-s + (−0.104 − 0.994i)4-s + (0.309 + 0.951i)6-s + (−0.978 + 0.207i)7-s + (0.809 + 0.587i)8-s + (−0.5 − 0.866i)9-s + (−0.669 + 0.743i)11-s + (−0.913 − 0.406i)12-s + (−0.104 − 0.994i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)16-s + (−0.809 + 0.587i)17-s + (0.978 + 0.207i)18-s + (0.104 − 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.249 + 0.968i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.249 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1011183656 + 0.1304306205i\)
\(L(\frac12)\) \(\approx\) \(0.1011183656 + 0.1304306205i\)
\(L(1)\) \(\approx\) \(0.6110480480 - 0.05473373891i\)
\(L(1)\) \(\approx\) \(0.6110480480 - 0.05473373891i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.669 + 0.743i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (-0.104 - 0.994i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.913 + 0.406i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.669 + 0.743i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.42718834852190167634984584231, −16.74923956080073582353260440811, −16.259220200614121199540494246548, −15.858617451102099042326627522921, −15.058989715375397983224651584716, −14.013539349817755907608587677560, −13.47421536289912357149874364648, −13.105658197285218870235700893806, −11.99036730672377388945246482432, −11.42601045560927769326603381547, −10.798822712282615681718591106949, −10.00682253308443439401746917648, −9.71760205541910160327609407300, −8.92638857979370482101496660473, −8.521485821756622825561764161008, −7.58770181716181649926516560439, −7.00889682889123847655085843703, −5.96736665635676258784999273345, −5.10886761242529200621505339270, −4.09534923913858596972700954506, −3.74778422374751552880296658005, −2.89909327455072380528418596879, −2.41550982832608964186935686884, −1.40899001029278295829570688909, −0.06775772384260676585999278544, 0.68018821102700377152045789533, 1.82818201587168984494806216421, 2.50124948173917045673908239219, 3.17652082442346942338987833478, 4.35235634245939718778994597480, 5.25549032210717048284077877147, 5.97492291325839454264208251654, 6.6537708246455340230958277607, 7.1575472307291298928400423897, 7.797363061783105996945478860785, 8.46836840288329615013206899117, 9.21829357332434128197123521606, 9.55063014796686607726277398220, 10.63696997931690377493612179263, 10.94833861253170188709335125556, 12.28067100633679483725954120731, 12.882786849995378486647757115650, 13.16019766545397572405771046681, 14.031192500196237334637465317345, 14.94590864453300193409616810586, 15.17684362060259559444722164160, 15.892860912665112498971226412079, 16.61119861304730267041175814706, 17.4830345424163948704179840692, 17.91021289647825942295884115644

Graph of the $Z$-function along the critical line